Martin Davis asked what the Foundations of Mathematics are for.
I present my opinion on these matters.
The following picture is an overly simple ordering of several scientific disciplines from
concrete to more abstract:
Biology < Chemistry < Physics < Mathematics < Mathematical Logic
Everyone will agree that Biology is not just the study of some particular
chemical reactions. Similarly, Chemistry (resp. Physics) is not just the study of some
particular physical (resp. mathematical) systems.
In each case, if X<Y, then X cannot be *reduced* to some subset of Y. Obviously
X does take place in a subset of Y, but it is more than just a subset of Y, something
"holistic" is going on.
A good example is Biology: the notion of "living matter" is not well-defined or understood
yet, and has no reduction (to the best of my knowledge) to simple chemical reactions.
Extrapolating, in the case of Mathematics, we cannot just reduce Mathematics to the study
of some formal logical systems. Something more is going on. Evidence for this, in my opinion,
is provided by the results of Reverse Mathematics. One does not need to agree with the "Big Five"
thesis to admit that Reverse Mathematics reveals surprising properties of "ordinary Mathematics".
These properties (discussed at length on this list) do not have reductions to formal logic, to the best
of my knowledge.
Hence, *a* role for FOM is the study of the properties of Mathematics that differentiate (formalized) Mathematics
from generic logical systems. In other words, not all logical systems have mathematical content and what
makes those that do (e.g. ACA_0, WKL_0, …) different?
Best,
Sam Sanders